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Mirrors > Home > MPE Home > Th. List > tpid2 | Structured version Visualization version GIF version |
Description: One of the three elements of an unordered triple. (Contributed by NM, 7-Apr-1994.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) |
Ref | Expression |
---|---|
tpid2.1 | ⊢ 𝐵 ∈ V |
Ref | Expression |
---|---|
tpid2 | ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2818 | . . 3 ⊢ 𝐵 = 𝐵 | |
2 | 1 | 3mix2i 1326 | . 2 ⊢ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶) |
3 | tpid2.1 | . . 3 ⊢ 𝐵 ∈ V | |
4 | 3 | eltp 4618 | . 2 ⊢ (𝐵 ∈ {𝐴, 𝐵, 𝐶} ↔ (𝐵 = 𝐴 ∨ 𝐵 = 𝐵 ∨ 𝐵 = 𝐶)) |
5 | 2, 4 | mpbir 232 | 1 ⊢ 𝐵 ∈ {𝐴, 𝐵, 𝐶} |
Colors of variables: wff setvar class |
Syntax hints: ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 Vcvv 3492 {ctp 4561 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-v 3494 df-un 3938 df-sn 4558 df-pr 4560 df-tp 4562 |
This theorem is referenced by: wrdl3s3 14314 wwlks2onv 27659 elwwlks2ons3im 27660 umgrwwlks2on 27663 s3rn 30549 cyc3evpm 30719 sgnsf 30731 sgncl 31695 signsw0glem 31722 signsw0g 31725 signswmnd 31726 signswrid 31727 prodfzo03 31773 circlevma 31812 circlemethhgt 31813 hgt750lemg 31824 hgt750lemb 31826 hgt750lema 31827 hgt750leme 31828 tgoldbachgtde 31830 tgoldbachgt 31833 kur14lem7 32356 brtpid2 32849 rabren3dioph 39290 fourierdlem102 42370 fourierdlem114 42382 etransclem48 42444 |
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